In a tangentially related post, Jeff confirmed that this odd date situation is related to deleted users and marked the bug status-deferred. This seems to confirm what Tim Stone found (see edit 1, below this).
As pointed out by Tim Stone in the chat, this is likely because the user who asked the question has been deleted (or for some other reason no longer exists).
I'm now seeing this on mouseover:
The question owner accepted this as the best answer Jan 1 '01 at 0:00
The completely expanded timeline view — perhaps unsurprisingly — don't show any indication of when the actual acceptance took place.
I don't see any indication of a bounty, including inside the revision history, though it's possible that it's being hidden by the same bug. Do you know for sure that there was one, or were you making an educated guess?
The very oldest CW question on SO with an accepted answer doesn't show this bug, so it's not an age thing.